The Dual Polynomial of Bipartite Perfect Matching

We obtain a description of the Boolean dual function of the Bipartite Perfect Matching decision problem, as a multilinear polynomial over the Reals. We show that in this polynomial, both the number of monomials and the magnitude of their coefficients are at most exponential in $\mathcal{O}(n \log n)$. As an application, we obtain a new upper bound of $\mathcal{O}(n^{1.5} \sqrt{\log n})$ on the approximate degree of the bipartite perfect matching function, improving the previous best known bound of $\mathcal{O}(n^{1.75})$. We deduce that, beyond a $\mathcal{O}(\sqrt{\log n})$ factor, the polynomial method cannot be used to improve the lower bound on the bounded-error quantum query complexity of bipartite perfect matching

Prompt Gamma Ray Burst emission from gradual magnetic dissipation

We considered a model for the prompt phase of Gamma-Ray Burst (GRB) emission arising from a magnetized jet undergoing gradual energy dissipation due to magnetic reconnection. The dissipated magnetic energy is translated to bulk kinetic energy and to acceleration of particles. The energy in these particles is released via synchrotron radiation as they gyrate around the strong magnetic fields in the jet. At small radii, the optical depth is large, and the radiation is reprocessed through Comptonization into a narrow, strongly peaked, component. At larger distances the optical depth becomes small and radiation escapes the jet with a non-thermal distribution. The obtained spectra typically peak around $\approx 300$keV (as observed) and with spectral indices below and above the peak that are, for a broad range of the model parameters, close to the observed values. The small radius of dissipation causes the emission to become self absorbed at a few keV and can sufficiently suppress the optical and X-ray fluxes within the limits required by observations (Beniamini & Piran 2014).Comment: 11 pages, 5 figures. Published in MNRA

What can we learn from "internal plateaus"? The peculiar afterglow of GRB 070110

Context: The origin of GRBs' prompt emission is highly debated. Proposed scenarios involve dissipation processes above or below the photosphere of an ultra-relativistic outflow. Aims: We search for observational features that would favour one scenario over the others by constraining the dissipation radius, the outflow magnetization or by indicating the presence of shocks. Bursts showing peculiarities can emphasize the role of a specific physical ingredient, which becomes more apparent under certain circumstances. Methods: We study GRB 070110, which exhibited several remarkable features during its early afterglow: a very flat plateau terminated by an extremely steep drop and immediately followed by a bump. We model the plateau as photospheric emission from a long lasting outflow of moderate Lorentz factor ($\Gamma\sim 20$) which lags behind an ultra-relativistic ($\Gamma> 100$) ejecta responsible for the prompt emission. We compute the dissipation of energy in the forward and reverse shocks resulting from this ejecta's deceleration by the external medium. Results: Photospheric emission from the long-lasting outflow can account for the plateau properties (luminosity and spectrum) assuming some dissipation takes place in the flow. The geometrical timescale at the photospheric radius is so short that the observed decline at the end of the plateau likely corresponds to the shut-down of the central engine. The following bump results from dissipated power in the reverse shock, which develops when the slower material catches up with the initially fast component, after it had been decelerated. Conclusions: Our interpretation suggests that the prompt phase resulted from dissipation above the photosphere while the plateau had a photospheric origin. If the bump is produced by the reverse shock, it implies an upper limit ($\sigma \lesssim 0.1$) on the magnetization of the slower material.Comment: 10 pages, 4 figures, accepted for publication in A&